Relationship between the focal length and radius of curvature of a spherical mirror

Consider a ray parallel to the principal axis strinking the mirror at M, CM will be perpendicular to the mirror at M.

Let θ be the angle of incidence and MQ be the perpendicular from M on the principal axis, then

∠MCP = θ and ∠MFP =2θ

Now

tanθ =MQ/CQ and tan2θ = MQ/FQ

for small θ, which is true for paraxial rays tanθ is aprox equal to θ and tan2θ is aprox equal to 2θ

Therefore from the above equation.

MQ/FQ = 2MQ/CQ

or FQ = CQ/2

Now, for small θ the point Q is very close to the point P therefore

FQ = f, CQ = R

Then from equation FQ = CQ/2

f = R/2

where f is focal length and R is radius of curvature.

Thus, In spherical mirror ( both concave and convex) focal length is equal to half of the radius of curvature.